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Lê cycles and hypersurface singularities. (English) Zbl 0835.32002

Lecture Notes in Mathematics. 1615. Berlin: Springer-Verlag. xi, 131 p. (1995).
Let \(f : (\mathbb{C}^{n + 1}, 0) \to (\mathbb{C}, 0)\) define a hypersurface singularity with \(s\)-dimensional singular locus. Using polar varieties the author introduces Lê-varieties \(\Lambda_f^{(k)}\), \(k = 0, \dots, s\), with the aim to obtain similar information in the case \(s > 0\) as in the case \(s = 0\) from the jacobian ideal. The multiplicities \(\text{mult} (\Lambda_f^{ (k)}) = : \lambda_f^{(k)}\) generalize the Milnor number: in case of an isolated singularity \( \Lambda_f^{(k)} = \emptyset\) for \(k > 0\), \(\Lambda_f^{(0)}\) is defined by the jacobian ideal, i.e. \(\lambda_f^{(0)}\) is the Milnor number.
It is proved that for \(s \leq n - 2\) the Milnor fibre of \(f\) is obtained from \(B^{2n}\) by successively attaching \(\lambda_f^{(n - k)}\) \(k\)- handles, \(n - s \leq k \leq n\).
If \(s = n - 1\) the Milnor fibre is obtained from a \(2n\)-manifold with homotopy type of a bouquet of \(\lambda_f^{(n - 1)}\) circles by successively attaching \(\lambda_f^{(n - k)}\) \(k\)-handles, \(2 \leq k \leq n\).
The author investigates the Milnor fibration of \(f\) using his Lê- varieties to extend the result of Lê and Ramanujam concerning the constancy of Milnor fibration in a family of isolated singularities to nonisolated singularities: the Lê numbers constant in a family imply constant homotopy type of the Milnor fibre.
This lecture notes tries to summarize all results known concerning Lê numbers. It starts with an introduction explaining the results of Milnor, Lê, Ramanujam and Saito concerning this subject, followed by two chapters with definitions and examples. Chapter 3 contains the structure of the Milnor fibre mentioned above. The constancy of the Milnor fibration is discussed in Chapter 9. Inbetween generalized Lê-Iomdine formulas, Lê numbers and hyperplane arrangements, Thoms \(a_f\) condition, aligned singularities and suspending singularities are discussed.

MSC:

32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces
32S55 Milnor fibration; relations with knot theory
32S25 Complex surface and hypersurface singularities
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